To prove Theorem 1.1, we first give a lemma.

Lemma 2.1

Let (fin W_{infty }^{n}). Assume that (Delta := {ale x_{1}< x_{2}<cdots <x_{r}le b,alpha _{i}in mathbb{N},n=sum_{i=1}^{r}alpha _{i}}) is a Hermite interpolation system. Then, the remainder (R_{Delta }(f,x):=f(x)-H_{Delta }(f,x)) for the Hermite interpolation polynomial based on Δ satisfies

$$ biglvert R_{Delta }(f,x) bigrvert = biglvert f(x)-H_{Delta }(f,x) bigrvert le frac{ Vert f^{(n)} Vert _{infty }}{n!} biglvert W_{Delta }(x) bigrvert ,quad xin [a,b], $$

(2.1)

where (W_{Delta }) is given by (1.4). In particular, if (fin C^{n}), then

$$ R_{Delta }(f,x)=f(x)-H_{Delta }(f,x)= frac{f^{(n)}(xi )}{n!}W_{ Delta }(x),quad xin [a,b], $$

(2.2)

for some (xi in [-1,1]) depending on x and Δ.

Proof

Since (2.1) is trivially satisfied if x coincides with one of the interpolation points (x_{1},ldots ,x_{r}), we need be concerned only with the case where x does not coincide with one of the interpolation nodes. Keeping x fixed, consider (g:[a,b]to mathbb{R}) given by

$$ g(y):=R_{Delta }(f,y)-W_{Delta }(y) frac{R_{Delta }(f,x)}{W_{Delta }(x)},quad yin [a,b]. $$

(2.3)

By the assumption on f we know (gin W_{infty }^{n}). From (1.3) and (2.3) we conclude that g has at least (n+1) zeros (counting multiplicity), namely single zero x and (alpha _{k}) fold zeros (x_{k},k=1,ldots ,r). Then, by Rolle’s theorem, the derivative (g’) has at least n zeros. Repeating the argument, by induction we deduce that the derivative (g^{(n-1)}) has at least two zeros in ([a,b]), which we denote by (z_{1}) and (z_{2}) ((z_{1}< z_{2})), respectively. Since (gin W_{infty }^{n}), then by the Newton–Leibniz formula we obtain

$$ 0=g^{(n-1)}(z_{2})-g^{(n-1)}(z_{1})= int _{z_{1}}^{z_{2}}g^{(n)}(y),dy. $$

(2.4)

It is known that (H_{Delta }(f)) is an algebraic polynomial of degree at most (n-1). Hence, we obtain

$$ bigl(H_{Delta }(f)bigr)^{(n)}(y)=0. $$

(2.5)

By a direct computation we obtain

$$ (W_{Delta })^{(n)}(y)=n!. $$

(2.6)

Substituting (2.5) and (2.6) into (2.4), we obtain

$$ 0= int _{z_{1}}^{z_{2}} biggl[f^{(n)}(y)-n! frac{R_{Delta }(f,x)}{W_{Delta }(x)} biggr],dy= int _{z_{1}}^{z_{2}}f^{(n)}(y),dy-n!(z_{2}-z_{1}) frac{R_{Delta }(f,x)}{W_{Delta }(x)}. $$

(2.7)

From (2.7) it follows that

$$ R_{Delta }(f,x)= frac{int _{z_{1}}^{z_{2}}f^{(n)}(y),dy}{n!(z_{2}-z_{1})}W_{Delta }(x). $$

(2.8)

Combining

$$ bigglvert int _{z_{1}}^{z_{2}}f^{(n)}(y),dy biggrvert le int _{z_{1}}^{z_{2}} biglvert f^{(n)}(y) bigrvert ,dy le int _{z_{1}}^{z_{2}} biglVert f^{(n)} bigrVert _{infty },dy= biglVert f^{(n)} bigrVert _{infty }(z_{2}-z_{1}) $$

with (2.8) we obtain (2.1). Besides, if (fin C^{n}), then (g^{(n)}) has at least one zero ξ in ([a,b]), i.e., (g^{(n)}(xi )=0). Hence, by differentiating n times on two sides of (2.3) first, and then substituting (2.5) and (2.6) into the obtained relation, we obtain

$$ 0=f^{(n)}(xi )-n!frac{R_{Delta }(f,x)}{W_{Delta }(x)}. $$

(2.9)

From (2.9) we obtain (2.2). This completes the proof of Lemma 2.1. □

Lemma 2.2

Let (1le p< infty ) and assume that (omega (x)> 0) is continuousintegrable on ((-1,1)). Then, there exists a unique (W_{n,p,omega }in mathcal{P}_{n} ) for all (nin mathbb{N}) such that

$$ Vert W_{n,p,omega } Vert _{p,omega }=E_{n,p,omega }quad textit{and}quad W_{n,p, omega }(x)=x^{n}+c_{1}x^{n-1}+ cdots +c_{n}, $$

where (E_{n,p,omega }) is given by (1.5). Furthermore, (W_{n,p,omega }) has exactly n zeros given by (1.7).

Proof

The proof of the problem on ([-1,1]) can be found in [16]. In general, we can use the variable substitution (x=frac{a+b}{2}+frac{b-a}{2}t) to refer the problem on ([a,b]) to this on ([-1,1]). We omit the details. □

Proof of Theorem 1.1

We consider (1) first. Let (Delta _{n,infty }) be given by (1.12). Then, for any (fin W_{infty }^{n}), it follows from (2.1) that

$$ begin{aligned}[b] &biglvert f(x)-L_{Delta _{n,infty }}(f,x) bigrvert le frac{ Vert f^{(n)} Vert _{infty }}{n!} Bigglvert prod_{i=1}^{n} biggl(x-frac{a+b}{2}-frac{b-a}{2}cos frac{(2i-1)pi }{2n} biggr) Biggrvert , \ &quad xin [a,b]. end{aligned} $$

(2.10)

Let (x=frac{a+b}{2}+frac{b-a}{2}t). Then, (2.10) becomes

$$begin{aligned} biglvert f(x)-L_{Delta _{n,infty }}(f,x) bigrvert le & frac{ Vert f^{(n)} Vert _{infty }(b-a)^{n}}{n!2^{n}} Bigglvert prod _{i=1}^{n} biggl(t-cos frac{(2i-1)pi }{2n} biggr) Biggrvert \ =&frac{ Vert f^{(n)} Vert _{infty }(b-a)^{n}}{n!2^{2n-1}} biglvert T_{n}(t) bigrvert , quad t in [-1,1], end{aligned}$$

(2.11)

where (T_{n}) is the nth Chebyshev polynomial of the first kind, i.e., (T_{n}(t)=cos (narccos t)). Let (fin BW_{infty }^{n}). Then, we have (|f^{(n)}|_{infty }le 1). Combining this fact with (|T_{n}|_{infty }=1) as well as (2.11), we obtain

$$ ebigl(BW_{infty }^{n},L_{Delta _{n,infty }},L_{infty } bigr)=sup_{fin BW_{infty }^{n}} biglVert f-L_{Delta _{n,infty }}(f) bigrVert _{infty }le frac{(b-a)^{n}}{n!2^{2n-1}}. $$

(2.12)

From (1.2) and (2.12) we obtain the upper estimate.

Now, we consider the lower estimate. Let (Delta := {ale x_{1}< x_{2}<cdots <x_{r}le b,alpha _{i}in mathbb{N},n=sum_{i=1}^{r}alpha _{i}}) be an arbitrary Hermite interpolation system of cardinality n in ([a,b]). Consider the function (g(x)=frac{x^{n}}{n!}). Then, from (g^{(n)}(x)=1) and (2.2) it follows that (gin W_{infty }^{n}) and

$$ g(x)-H_{Delta }(g,x)=frac{W_{Delta }(x)}{n!},quad xin [a,b]. $$

(2.13)

Let (x=frac{a+b}{2}+frac{b-a}{2}t). Then, by (1.4) we obtain

$$ W_{Delta }(x)=x^{n}+a_{1}x^{n-1}+a_{2}x^{n-2}+ cdots +a_{n}= frac{(b-a)^{n}}{2^{n}}h(t),quad tin [-1,1],$$

(2.14)

where

$$ h(t)=t^{n}+b_{1}t^{n-1}+b_{2}t^{n-2}+ cdots +b_{n}. $$

(2.15)

Then, it follows from Theorem 6.1 in [4, Ch. 3] that

$$ Vert h Vert _{infty }ge 2^{1-n}.$$

(2.16)

Combining (1.1), (2.13), (2.14) and (2.16), we obtain

$$ ebigl(BW_{infty }^{n}, H_{Delta }, L_{infty }bigr)ge biglVert g-H_{Delta }(g) bigrVert _{infty }=frac{ Vert W_{Delta } Vert _{infty }}{n!}=frac{(b-a)^{n}}{n!2^{n}} Vert h Vert _{infty }ge frac{(b-a)^{n}}{n!2^{2n-1}}.$$

(2.17)

From (1.2) and (2.17) we obtain the lower estimate.

Next, we consider (2). We consider the upper estimate first. Let (Delta _{n,p,omega }) be given by (1.8) and (W_{n,p,omega }) be given by (1.6). If (fin BW_{infty }^{n}), then we have (|f^{(n)}|_{infty }le 1). Combining this fact with (2.1) we obtain

$$ biglvert f(x)-L_{Delta _{n,p,omega }}(f,x) bigrvert le frac{ vert W_{n,p,omega }(x) vert }{n!}, quad xin [a,b]. $$

It follows that

$$ biglVert f-L_{Delta _{n,p,omega }}(f) bigrVert _{p,omega }le frac{ Vert W_{n,p,omega } Vert _{p,omega }}{n!}= frac{E_{n,p,omega }}{n!}. $$

(2.18)

From (1.1) and (2.18) we obtain

$$ ebigl(BW_{infty }^{n},L_{Delta _{n,p,omega }},L_{p,omega } bigr)le frac{E_{n,p,omega }}{n!}. $$

(2.19)

From (1.2) and (2.19) we obtain the upper estimate.

Now, we consider the lower estimate. Let (Delta := {ale x_{1}< x_{2}<cdots <x_{r}le b,alpha _{i}in mathbb{N},n=sum_{i=1}^{r}alpha _{i}}) be an arbitrary Hermite interpolation system of cardinality n in ([a,b]). Consider the function (g(x)=frac{x^{n}}{n!}). Then, (gin W_{infty }^{n}) and (2.13) holds. From the first equality in (2.14) and (1.5) as well as (1.6) it follows that

$$ Vert W_{Delta } Vert _{p,omega }ge E_{n,p,omega }. $$

(2.20)

From (1.1), (2.13) and (2.20) it follows that

$$ ebigl(BW_{infty }^{n}, H_{Delta }, L_{p,omega }bigr)ge biglVert g-H_{Delta }(g) bigrVert _{p, omega }=frac{ Vert W_{Delta } Vert _{p,omega }}{n!}ge frac{E_{n,p,omega }}{n!}.$$

(2.21)

From (1.2) and (2.21) we obtain the lower estimate of (2). Theorem 1.1 is proved. □

Let (BC^{n}={fin C^{n}:|f^{(n)}|_{infty }le 1}). Using the fact (BW_{infty }^{n}subset BC^{n}) and (g(x)=frac{x^{n}}{n!}in BC^{n}) for (nin mathbb{N}), combining the proof of Theorem 1.1, we obtained the following results.

Corollary 2.3

  1. (1)

    For (p= infty ), we have

    $$ ebigl(n,BC^{n} ,L_{infty }bigr)=ebigl(BC^{n},L_{Delta _{n,infty }}, L_{infty }bigr)= frac{(b-a)^{n}}{n!2^{2n-1}},$$

    (2.22)

    where (Delta _{n,infty }) is given by (1.12).

  2. (2)

    Let (1le p< infty ) and assume that (omega (x)> 0) is continuousintegrable on ((a,b)). Then, we have

    $$ ebigl(n,BC^{n} , L_{p,omega }bigr)=ebigl(BC^{n},L_{Delta _{n,p,omega }}, L_{p, omega }bigr)= frac{E_{n,p,omega }}{n!},$$

    (2.23)

    where (Delta _{n,p,omega }) is given by (1.8).

Proof of Theorem 1.2

We consider (1) first. For any (fin BW_{infty }^{n}), from (2.1) it follows that

$$ begin{aligned}[b] &biglvert f(x)-L_{Delta ^{*}_{n,infty }}(f,x) bigrvert le frac{1}{n!} Bigglvert prod_{i=1}^{n} biggl(x-frac{a+b}{2}-frac{b-a}{2}cos frac{(2i-1)pi }{2n} Big/ cos frac{pi }{2n} biggr) Biggrvert , \ &quad xin [a,b]. end{aligned} $$

(2.24)

Let (x=frac{a+b}{2}+frac{b-a}{2cos frac{pi }{2n}}t). Then, we have

$$ begin{aligned}[b] &prod_{i=1}^{n} biggl(x-frac{a+b}{2}-frac{b-a}{2}cos frac{(2i-1)pi }{2n} Big/ cos frac{pi }{2n} biggr)= frac{(b-a)^{n}T_{n}(t)}{ (cos frac{pi }{2n} )^{n}2^{2n-1}}, \ &quad tin biggl[-cos frac{pi }{2n},cos frac{pi }{2n} biggr]. end{aligned} $$

(2.25)

From (1.1), (2.24) and (2.25) it follows that

$$ begin{aligned}[b] ebigl(BW_{infty }^{n}, L_{Delta ^{*}_{n,infty }}, L_{infty }bigr)&le frac{(b-a)^{n}}{ (cos frac{pi }{2n} )^{n}2^{2n-1}n!} sup_{tin [-cos frac{pi }{2n},cos frac{pi }{2n} ]} biglvert T_{n}(t) bigrvert \ & = frac{(b-a)^{n}}{ (cos frac{pi }{2n} )^{n}2^{2n-1}n!}. end{aligned} $$

(2.26)

From (1.2) and (2.26) we obtain the upper estimate.

Now, we consider the lower estimate. Let (Delta := {a= x_{1}< x_{2}<cdots <x_{r}= b,alpha _{i}in mathbb{N},n=sum_{i=1}^{r}alpha _{i}}) be an arbitrary Hermite interpolation system of cardinality n including the endpoints a and b. Consider the function (g(x)=frac{x^{n}}{n!}). Then, (gin W_{infty }^{n}) and (2.13) holds. Let (x=frac{a+b}{2}+frac{b-a}{2}t). Denote (t_{i}=frac{2}{b-a} (x_{i}-frac{a+b}{2} )), (i=1,ldots ,r). Then, by (1.4) one obtains

$$ W_{Delta }(x)=frac{(b-a)^{n}}{2^{n}}prod _{i=1}^{r}(t-t_{i})^{ alpha _{i}}, quad t_{1}=-1,, t_{r}=1,, tin [-1,1].$$

(2.27)

Let

$$ g(t)=bigl(t^{2}-1bigr)prod_{i=2}^{n-1} biggl(t-cos frac{(2i-1)pi }{2n} Big/cos frac{pi }{2n} biggr)= frac{T_{n}(tcos frac{pi }{2n})}{2^{n-1} (cos frac{pi }{2n} )^{n}}. $$

Then, it is easy to verify that

$$ Vert g Vert _{infty }= frac{1}{2^{n-1} (cos frac{pi }{2n} )^{n}} $$

(2.28)

and

$$ g biggl(frac{cos frac{ipi }{n}}{cos frac{pi }{2n}} biggr)= frac{(-1)^{i}}{2^{n-1} (cos frac{pi }{2n} )^{n}},quad i=1, ldots ,n-1. $$

(2.29)

Assume that

$$ BigglVert prod_{i=1}^{r}(t-t_{i})^{alpha _{i}} BiggrVert _{infty }< frac{1}{2^{n-1} (cos frac{pi }{2n} )^{n}}. $$

(2.30)

Let

$$ R(t)=g(t)-prod_{i=1}^{r}(t-t_{i})^{alpha _{i}}, quad tin [-1,1]. $$

Then, it is easy to verify that (R(t)) is a polynomial of degree at most (n-1). Furthermore, from (2.29) and (2.30) one can check that

$$ R biggl(frac{cos frac{ipi }{n}}{cos frac{pi }{2n}} biggr) (-1)^{i}>0,quad i=1, ldots ,n-1. $$

Thus, the polynomial (R(t)) has at least (n-2) zeros in ((-1,1)). As (t_{1}=-1), (t_{r}=1), it is clear that ±1 are zeros of (R(t)). Hence, (R(t)) has at least n zeros in ([-1,1]). This, and the fact that (R(t)) is a polynomial of degree at most (n-1), implies that (R(t)=0). Therefore,

$$ BigglVert prod_{i=1}^{r}(t-t_{i})^{alpha _{i}} BiggrVert _{infty }= Vert g Vert _{infty }= frac{1}{2^{n-1} (cos frac{pi }{2n} )^{n}}, $$

which contradicts (2.30). Hence, we have

$$ BigglVert prod_{i=1}^{r}(t-t_{i})^{alpha _{i}} BiggrVert _{infty }ge frac{1}{2^{n-1} (cos frac{pi }{2n} )^{n}}. $$

(2.31)

From (1.1), (2.13), (2.27) and (2.31) we obtain

$$ ebigl(BW_{infty }^{n}, H_{Delta }, L_{infty }bigr)ge biglVert g-H_{Delta }(g) bigrVert _{infty }=frac{ Vert W_{Delta } Vert _{infty }}{n!}ge frac{(b-a)^{n}}{ (cos frac{pi }{2n} )^{n}2^{2n-1}n!}.$$

(2.32)

From (1.2) and (2.32) we obtain the lower estimate of (1).

Next, we consider (2). Let ω̅ and (Delta ^{*}_{n,p,omega }) be given by (1.19). Then, for any (fin BW_{infty }^{n}), from (2.1) it follows that

$$ biglvert f(x)-L_{Delta ^{*}_{n,p,omega }}(f,x) bigrvert le frac{(1-x^{2}) vert W_{n-2,p,overline{omega }}(x) vert }{n!},quad xin [a,b]. $$

(2.33)

From (2.33) it follows that

$$ biglVert f-L_{Delta ^{*}_{n,p,omega }}(f) bigrVert _{p,omega }le frac{ Vert W_{n-2,p,overline{omega }} Vert _{p,overline{omega }}}{n!}= frac{E_{n-2,p,overline{omega }}}{n!}. $$

(2.34)

From (1.1) and (2.34) we conclude that

$$ ebigl(BW_{infty }^{n},L_{Delta ^{*}_{n,p,omega }},L_{p,omega } bigr)le frac{E_{n-2,p,overline{omega }}}{n!}. $$

(2.35)

On the other hand, let (Delta := {a= x_{1}< x_{2}<cdots <x_{r}= b,alpha _{i}in mathbb{N},n=sum_{i=1}^{r}alpha _{i}}) be an arbitrary Hermite interpolation system of cardinality n including the endpoints. Consider the function (g(x)=frac{x^{n}}{n!}). Then, (gin W_{infty }^{n}) and (2.13) holds. From (1.1), (2.13), (1.5) and (1.6) it follows that

$$begin{aligned} ebigl(BW_{infty }^{n}, H_{Delta }, L_{p,omega }bigr) ge & biglVert g-H_{Delta }(g) bigrVert _{p, omega }= frac{1}{n!} BigglVert prod_{k=1}^{r}(x-x_{k})^{alpha _{k}} BiggrVert _{p,omega } \ =& frac{1}{n!} BigglVert (x-a)^{alpha _{1}-1}(b-x)^{alpha _{r}-1} prod_{k=2}^{r-1}(x-x_{k})^{alpha _{k}} BiggrVert _{p,overline{omega }} \ ge & frac{1}{n!} BigglVert prod_{k=1}^{n-2}(x- xi _{k,p, overline{omega }}) BiggrVert _{p,overline{omega }}= frac{E_{n-2,p,overline{omega }}}{n!}. end{aligned}$$

(2.36)

From (2.35) and (2.36) as well as (1.2) we obtain the result of (2). Theorem 1.2 is proved. □

Using the fact that (BC^{n}subset BW_{infty }^{n}) and (g(x)=frac{x^{n}}{n!}in BC^{n}) for (nin mathbb{N}), combining the proof of Theorem 1.2, we obtained the following results.

Corollary 2.4

  1. (1)

    Let (p= infty ) and (n>2). Then, we have

    $$ overline{e}bigl(n,BC^{n}, L_{infty }bigr) =e bigl(BC^{n}, L_{{Delta ^{*}_{n, infty }}}, L_{infty }bigr)= frac{(b-a)^{n}}{ (cos frac{pi }{2n} )^{n}2^{2n-1}n!},$$

    where (Delta ^{*}_{n,infty }) is given by (1.17).

  2. (2)

    Let (1le p< infty ), (n>2) and assume that (omega (x)> 0) is continuousintegrable on ((a,b)). Then, we have

    $$ overline{e}bigl(n,BC^{n}, L_{p,omega }bigr) =e bigl(BC^{n}, L_{Delta ^{*}_{n,p, omega }}, L_{p,omega }bigr)= frac{E_{n-2,p,overline{omega }}}{n!},$$

    where ω̅ and (Delta ^{*}_{n,p,omega }) are given by (1.19).

Remark 2.5

When (nne r), the nth optimal Hermite interpolation system of the problems given by (1.2) and (1.15) for (BW_{infty }^{r}) in (L_{infty }) and (L_{p,omega }) ((1le p<infty )) are open problems.

Remark 2.6

When (n= r), the nth optimal Birkhoff interpolation system of the problems given by (1.2) and (1.15) for (BW_{infty }^{n}) in (L_{infty }) and (L_{p,omega }) ((1le p<infty )) are open problems.

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